To me, it can mean only one thing, that Nixon really was like the NY Times liberal columnist, meaning he was a liberal. And if you look at Nixon's policies, he was a big government guy through and through. The only policy difference with the liberals, was that Nixon was an anti-communist, while liberals just considered communists to be "liberals in a hurry". Another phrase from the day was that "there were no enemies to the left". Every conservative or even moderate could be considered a political enemy, but not a socialist, marxist, communist or other statist.
Nixon engendered the hatred of the elitist liberals back when he was outing communists in the government and went after Alger Hiss, the epitomy of "one of us". Nixon was also declasse, a nobody with a nothing background and definitely not an alum of one of the right schools.
What else could Wicker mean by "one of us"?
It sounds pretty shallow and tinny if it's just a statement about ideology. That may be part of it, but I think it's more this:
What Wicker sees in Nixon--and what he believes the American public saw--is "one of us." Forget that John F. Kennedy beat Nixon in 1960. For Kennedy--the handsome, charming millionaire's son--beating Nixon should have been easy. Kennedy was the nation's "romantic dream of itself." That Nixon, who represented a "harder and clearer national self-assessment," came so close to winning (and in fact probably did, but for some vote-rigging in Chicago) is the real story, Wicker suggests.
In Nixon, the middle-class son of a saintly mother and a loud, nasty, Black Irish father, Americans saw "themselves as they knew they were . . . working and scheming without let to achieve their dreams, soured by the inequities of life." Asks Wicker: "Which of us in the national rush to get ahead has never cut a corner or winked at the law?" Businessweek
This sort of thing happens with reporters and politicians. After their fighting days are over, they can look at each other in a different light.
So sure, sometime in the 2020s a liberal reporter will write something like that about George W. Bush (not Paul Krugman, though, who's close to autistic). It may be that he sees some good in Bush's policies, but I think it's more that he sees that the human story is more interesting and maybe even bigger than the ideological one.